The hundred's digit of `3^(100)` is

Options

A. 0
B. 1
C. 2
D. 7

Answer

To find the hundred's digit of \(3^{100}\), we can follow these steps:

### Step 1: Express \(3^{100}\) in a manageable form
We start by rewriting \(3^{100}\) in terms of a base that is easier to work with. We know that \(3^2 = 9\), so we can express \(3^{100}\) as:
\[ 3^{100} = (3^2)^{50} = 9^{50} \]

### Step 2: Use the binomial theorem
Next, we can express \(9^{50}\) in a form that allows us to apply the binomial theorem. We can rewrite \(9\) as \(10 - 1\):
\[ 9^{50} = (10 - 1)^{50} \]
Now, we can expand this using the binomial theorem:
\[ (10 - 1)^{50} = \sum_{k=0}^{50} \binom{50}{k} 10^{50-k} (-1)^k \]

### Step 3: Identify the relevant terms
To find the hundred's digit, we need to focus on the terms that contribute to the \(10^2\) (hundred's place) in the expansion. This means we are interested in the terms where \(50 - k = 2\), or \(k = 48\):
\[ \text{Term for } k = 48: \binom{50}{48} 10^2 (-1)^{48} = \binom{50}{2} 10^2 \]

### Step 4: Calculate \(\binom{50}{2}\)
Now we calculate \(\binom{50}{2}\):
\[ \binom{50}{2} = \frac{50 \times 49}{2} = 1225 \]
Thus, the term for \(k = 48\) is:
\[ 1225 \times 10^2 = 122500 \]

### Step 5: Identify the hundred's digit
Next, we need to consider the term for \(k = 49\) (which contributes to the \(10^1\) place):
\[ \text{Term for } k = 49: \binom{50}{49} 10^1 (-1)^{49} = -\binom{50}{1} 10^1 = -50 \times 10 = -500 \]

### Step 6: Combine the contributions
Now, we combine the contributions from the terms we have calculated:
\[ 122500 - 500 = 122000 \]

### Step 7: Determine the hundred's digit
The hundred's digit of \(122000\) is \(0\).

### Final Answer
Thus, the hundred's digit of \(3^{100}\) is:
\[ \boxed{0} \]